This comparison graphic, together with the above caption, was
run as ad copy in a number of western Pennsylvania newspapers a couple of weeks
before the July 28, 2007 public meeting of the Flight 93 Memorial Project. If
your visit to this site was prompted by this ad, thanks for having eyes to see.
It is amazing how many people do not.

While the overt likeness between a traditional Islamic sundial
and the Tower of Voices can be seen just by looking, the shadow calculations
necessary to confirm that the Tower of voices is in fact an Islamic sundial are
quite tedious. In contrast, other Islamic and terrorist-memorializing features,
like the Mecca orientation of the giant crescent, are trivially easy to confirm.
Those looking for the simplest proofs of the worst features should begin with
either the Proof of Betrayal page, or with the Crescent
of Betrayal poster
(a high resolution map of the site plan with critical information superimposed).

If anyone actually wants to check the shadow calculations for
the tower-sundial, my
original calculations were posted on my
Error Theory blog in October 2005.
The presentation is cleaned up a bit in my Crescent of Betrayal book,
where Chapter 4
(in download 2) provides the basic
analysis. Detailed calculations appear in Addenda Seven and Eight, which for
convenience are reproduced below.

Important update:Memorial
Project Chairman John Reynolds dismisses my sundial analysis on grounds that
prove he has never even looked at it! (Muslim prayer times are determined by
shadow length, while Reynolds' argument assumes they are determined by shadow
angle.)

Unlike the tricky sundial in the final design that switches
gnomon height sometime in mid-November, the sundial in the
preliminary design is a simple traditional
Islamic sundial. It is on the same epic scale as the sundial in the final
design. Both are built around a similar height Tower of Voices (specified at 93
feet in the final design and seeming to be about the same in the original
design). But where the final design looks at first glance to be slightly
inaccurate between mid-November and late-January, the original is
immediately seen to give the proper time for Islamic afternoon prayers
throughout the entire year (assuming a 93 foot Tower height).

The original sundial was also in a completely different
location from the final one, so nobody can say this is not a completely
different sundial. A single year-round accurate Islamic prayer-time sundial
could never occur by accident. Two of them? Double never.

Addendum Seven: Shadow calculations and
other esoterica from the Tower-sundial

Here is my graph of where the tower's shadow falls at the
beginning of the time for Islamic afternoon prayers. As the year progresses from
June to December, the shadow hits further and further up the magenta
prayer-line, then starts moving back down.

To understand the zig zag in the prayer line, you have to
read Chapter 4. This is not the place to start trying to understand the Tower
sundial!

To put together my shadow calculations, I took advantage of
a number of very helpful online resources. Most important is J. Giesen’s Sun
Shadow Applet.[1]
Giesen’s applet the length and direction of the shadow cast by any gnomon, at
any time of the day, at any point on the earth. Just set the applet to 40.03
north latitude and 78.54 West longitude, set GMT to -5, turn off Daylight
Savings Time, and set the gnomon height to 28.35 meters (93 feet), which is
height of the Tower at the bottom of its crescent shaped top (the highest point
of solid shadow, whether or not the slot is in play). Since the bottom of the
crescent coincides with the slotted southwest side of the tower, it is this side
of the tower that serves as the origin from which shadow length and angle are
projected.

Giesen’s applet can be used to calculate Islamic afternoon
prayer times (asr prayers). For any day, use the applet to find the shortest
shadow of the day (somewhere near noon). Add this length to 28.35 meters, and
look for the time in the afternoon when the Tower’s shadow reaches this length.
That is the asr prayer time. Simpler is just to go to one of the Islamic
websites that hosts an Islamic prayer time calculator. I used the one at
qibla.org. Just plug in the latitude and longitude of the crash site, set GMT to
-5, turn Daylight Savings Time off, and the calculator will generate daily
prayer times by month and year.

With prayer time for a given day in hand, plug this time
and date into Giesen’s applet and it will tell you the length of the shadow and
the angle of the shadow. It gives the angle by giving the Sun’s azimuth in
degrees clockwise from north. Subtract 180 from this to get the angle of the
shadow clockwise from north. If your graphics program enables polar coordinates,
you can plot these lengths and angles directly onto a screen shot of the Tower
array. (On a Windows PC, pressing “Alt” and “Print Screen” will copy a screen
shot of your active browser window to you clipboard, allowing you to paste it
into other programs. To take a screen shot of your entire screen, omit the
“Alt”.)

If you are using Microsoft Paint you will need to do what I
did and convert your polar coordinates (angle and length) into Cartesian
coordinates. For this, I used the online coordinate converter provided by
Kusashi.com.[2]
This calculator assumes that angles are measured counter-clockwise from east, so
you will have to subtract your degrees clockwise from north from 90 before using
it. Plug this angle into the “argument” box, plug shadow length into the modulus
box and press calculate. The real component of the result will be your x
coordinate and the imaginary component of the result will be your y coordinate
(both in meters, since that was what you inputted in the modulus).

These meters then need to be converted to pixels, according
to the number of pixels to the meter in the screen-shot you are using. When I
take a screen shot of Murdoch’s Tower Plan PDF zoomed to 900%, a pixel count of
the scale shows 4.74 pixels to the meter. (If you want to download the original
PDF and verify this for yourself, the PDF is now behind password protection in
the Park Service archives You can get a password by calling 814 443-4557.) Thus
I multiply my meter measurements by 4.74 to plot them on my Tower Plan graphic.
It is a little cumbersome, but pretty simple once you get everything set up.

Here is the bit of screen shot on which I graphed my
results. (Click on image to bring up the original.)

And here are my shadow calculations, (slightly different
from the calculations in my original sundial post because I discovered that I
had entered the latitude wrong for my original calculations, 40.3 degrees north
instead of 40.03 degrees north):

The critical point
is the bottom of the gnomon, which as I constructed it has Cartesian coordinates
of (x,y) = (164, 426). (Coordinates are counted down to the right, starting
from the origin at the upper left. I chose this point by inscribing the crescent
shaped cross-section of the Tower with a circle and projecting a line through
the center of the circle that follows the angle of the Tower platform (rise 100,
run 70). Where this line cuts the bottom of the circle is the gnomon point. (The
color versions use (163, 426) as their origin point. (164, 426) looks more
accurate, so I switched to this for the B+W.)

Adding the June 16th run and rise of (177, -2)
to the (41, 324) gnomon location puts the end of the June 16th asr
prayer-shadow at (341, 428). [Note that Paint counts pixels from the upper left
corner of the graphic, so it reads (Δx, Δy) of (177, -2) as (177, 2).]

July 16th 2005
Asr = 4:19 PM; (modulus, argument) = (38.71, 1.8); (x, y) in meters = (38.70,
1.22); (x, y) in pixels = (183, 6), which is very close to the line for June,
and just a few pixels longer. The changes become more rapid as the sun gets
lower. Upper point at (347, 420). The Asr connecting line ending at this point
starts at (341, 428) and travels (Δx, Δy) = (6, -8). [Paint reads (Δx, Δy) = (6,
-8) as (Δx, Δy) = (7, -9) because it counts from one instead of zero. You can’t
draw a line zero px. long in Paint.]

November 16th 2005
For November and December, there are two shadow ends to calculate, one taking
the bottom of the top crescent to be the top of the tower and one taking the
bottom of the upper slot to be the top of the tower:

Full tower height:
Asr = 2.36 PM; (mod, arg) = (86.32 , 55.5); (x, y meters) = (48.89, 71.14); (x,
y pixels) = (232, 337). Upper point at (396, 89). The Asr connecting line
ending at this point starts at (398, 149) and travels (Δx, Δy) = (-2, -60).
[Microsoft Paint does not plot this line segment correctly, so in my graphic I
just placed a vertical line starting 1 pixels left of the previous Asr line
segment.]

Shadow from bottom of upper slot:
82% of (232, 337) is (190, 276). Lands at (355, 237). The Asr connecting line
ending at this point starts at (356, 199) and travels (Δx, Δy) = (-2, -49).

Dividing date between tall and short gnomon heights:

A line along the Tower shadow in Murdoch’s Tower Plan
detail has a slope of 194 pixels of rise for every 191 px. of run. That is a
little less than 45 degrees down from north, 44.55 degrees from N to be exact.

On November 3rd 2005, asr prayers occur at 2:47
PM, when the sun’s azimuth is 224.5 degrees from N, projecting its shadow to
44.5 degrees from north, almost exactly the angle depicted in Murdoch’s graphic.
Since light is reaching the ground through the slot at this angle (as can be
seen from the graphic), that means that the lower gnomon height, created by the
bottom of the slot, is what determines prayer time on November 3rd.

It appears from the graphic that the light through the slot
is about to get closed off as the sun continues westward. Note that the crescent
edge, visible at the tip of the Tower shadow, is lined up the thin shaft of
light making it through the slot. Any more westward movement of the sun and this
crescent edge will block light through the slot from reaching the ground. Since
prayers occur later at earlier dates, this means that light through the slot
probably does not reach the ground at prayer time just a couple of days earlier.
The exact date cannot be determined from the graphic so I used November 1st
as a best guess.

To plot the transition, it is necessary to calculate both
the tall and short prayer-time shadows for this date.

Starting from the Tower Detail gnomon position of (x, y) =
(163, 426), these tall and short shadow-falls land at (397, 195) and (355, 237)
respectively. These points demark the zig and the zag in the prayer-shadow-line,
which then connects the dots to the other shadow end points. The zig zag
segments are plotted as follows:

From the Oct. 16th Asr shadow to the upper point
of the zig, start at (396, 245) and travel (Δx, Δy) = (1, -50).

The zig line starts at (397, 195) and travels (Δx, Δy) =
(-42, 42).

From the bottom of the zag to the Nov. 16th Asr
shadow for the short gnomon height, start at (355, 237) and travel (Δx, Δy) =
(1, -38).

Fer de Vries computer program

Sundial expert Fer de Vries developed a computer program
for calculating Islamic prayer lines for any specified gnomon at any specified
latitude. I sent him an early version of my connect the dots derivation of the
Tower’s asr-prayer-line and he sent it back with his computer generated lines
for the specified gnomon height and the Shanksville latitude superimposed:

Fer de Vries’ asr prayer-line matches mine almost exactly.
The validity of the results still depend on the validity of my assumption that
the Tower base is at the same level as the trees, and Fer's program does not
address the validity of the zig zag in the asr line, but his program does
confirm that my calculations for the main asr line are reasonably accurate.
[3]

The migrating gnomon

For those who want to be completely thorough, note that
once light through the slot gets closed off by the westward movement of the sun,
the point that casts the bottom of the top of the Tower’s shadow will start to
move up the westward arm of the Tower’s crescent shaped top. The sunlight that
comes over the bottom of the crescent shaped top will, like any sunlight through
the slot, get blocked from reaching the ground by the eastern wall of the tower.
The two arms of the crescent will create a shadow with a pointed bottom (one
side vertical, one side curved out and up) that gets shallower in shape as the
sun sets.

Technically, the gnomon migrates clockwise around the
tower, starting at the slot and moving around to about due west. This will have
very little effect on where the bottom of the top of the Tower’s shadow lands at
asr-prayer time. Initially, the gnomon point comes around to the west (away from
the ellipse of trees) more than it rises. That will pull the asr-prayer line
inwards a bit in late October, where it is out towards the outer side of the
ellipse of trees, but it will not affect the tip of the zig.

Once the sun reaches due west (where it is at June asr
prayer times) the gnomon point will have climbed by about .4 of the radius of
the tower. (If the Tower top were sliced at a 45 degree angle and it was pointed
north, then the western side of the tower would be the taller than the slot side
by the radius of the Tower. As it is, the tower top is sliced at a 30 degree
angle, according to the Tower Section view, yielding 2/3s of the height gain of
a 45 degree slice. Also, the tower-crescent points 35 degrees clockwise from
north. That cuts the rise down by roughly 35/90 = 39 percent, and 61 percent of
2/3s = .41.) At 35 degrees from north, the slot-gnomon is positioned .57r west
of the center of the Tower circle (sine or cosine of 35, depending on where you
are measuring your degrees from). The due-west gnomon (the June gnomon), is r
west of center, so it moves west (away from the ellipse of trees) by about the
same amount that it rises: about .4r. Lastly, June asr prayer shadows descend at
an angle substantially flatter than 45 degrees (their length of 37.39 meters,
compared to the Tower height of 28.35 meters, implies an angle of descent of
36.22 degrees). With the June gnomon rising about the same amount as it moves
away from the tree line, this flatter than 45 degree shadow path to earth will
push the Tower’s asr-prayer time out by a modest fraction of the radius of the
tower. Since the prayer-line at that point is a bit towards the inside of the
tree line, this moves it a little closer to the base of the trees.

Addendum 8: Discrepancies between
different site-plan views in the tower-sundial analysis

Shadow calculations are sensitive to any change in
assumptions about the topography of the ground where the shadows are landing, or
about the exact dimensions of the shadow-caster. The calculations above require
a couple of assumptions, but they both seem to be well justified.

The first assumption is that the bottom of the Tower is at
the same level as the inner ellipse of trees (i.e. that the ground is flat). Is
this assumption correct? The site-plans include a Tower Section detail that
shows the Tower sitting on a raised base, about ten feet above grade. It also
depicts the trees in the background as planted at grade-level, below the raised
base. If this section view is correct then my assumptions about the topography
are incorrect. Instead of sitting at the same level as the trees, the Tower sits
ten feet above, and its shadows fall correspondingly further out.

Tower Section view from the Tower of Voices PDF page, shows
base of tower well above the grade where the trees are planted behind. If this
is accurate, the tower is a good ten feet higher than I was assuming, and its
prayer-time shadows will fall out beyond the inner arc of trees. [Med rez image.
More rez available in PDF.]

As it turns out, the Tower Section view misrepresents the
topography of the site. Terraserver’s topo map of the area shows that in the
fifty meters from the Tower to the tree line, the grade slopes up about ten
feet: just enough to bring the trees up to the bottom of the base of the Tower.
The map also shows a topo line running pretty much along the path of the asr
prayer line, meaning that the line of trees is itself level (not sloping up to
the left as depicted in the Tower Section view).

Terraserver’s topo map of Tower of Voices site. The Tower
is located just about at the last zero in 2400. There is twenty feet of
elevation change between contour lines, and the inner arc of trees is about
half-way to the next higher contour line to the right, making for an elevation
gain at the tree line of about ten feet.[1]

Is Murdoch’s Tower Section detail intentionally
misleading? Having done the shadow calculations himself, Murdoch would know that
there are sundial societies like NASS (the North American Sundial Society) with
experts on Islamic sundials who could recognize his sundial in a minute if he
didn’t disguise it.[2]
I sent my shadow calculations to Islamic-sundial expert Fer de Vries, who
quickly verified my asr line. (See Addendum 6 for Fer’s calculations.)
Thus it makes sense that Murdoch might want to misrepresent the topography of
the site so as to throw off the shadow calculations of anyone who tried to
investigate. What we can say for certain is that, intentionally or not, the
Tower Section view is misleading.

Sundial Assumption Two: The Depth of the Slot

A second assumption in my calculations is that the slot
depth indicated by the shadow in the Tower Plan drawing (18 percent of the Tower
height) is accurate. This information is contradicted, however, by the depth of
the slot depicted in the Tower Section view (above). Here the slot is claimed to
be very short, only 8 percent of the Tower height. So which is the real plan?
Short slot or long slot?

A third view is also available, the view up the tower, seen
in Figure 40 above. This view depicts the slot coming well down into the field
of wind-chimes. Contrast this with the Tower Section view, where the slot is
depicted as coming down only to very slightly below the top of the topmost wind
chime. It is hard to tell exactly how deep the slot is cut in the up-the-tower
view, but it is definitely a lot more than 8 percent (or one-twelfth) of the
way.

That makes it two views against one, and the view that
indicates the shorter slot has already been proven to contain misinformation
about the topography of the Tower site, giving another reason to favor the
information in the other two views.

Perhaps most importantly, note that the plan-view, which
shows the slot at 18% of the height of the Tower, is a CAD projection from a 3D
model. You can tell that by the shadows cast by the trees. A computer calculated
the shadows, implying that a computer also calculated the Tower shadow, based on
a 3D model of the Tower. The existence of such a 3D model is pretty strong
evidence that this is the real design. At the least, there is a lot more work
involved in creating a 3D model of the slot, and what we can certainly say is
that at least ONE of Murdoch's specifications creates a year round accurate
Islamic prayer-time sundial.

The Mecca Orientation

Finding the Islamic sundial adds meaning to the Mecca
orientation of the Tower crescents. When time for afternoon prayers arrives, the
Muslim faithful will need some way to determine the direction to Mecca, and
Murdoch has provided it. All they have to do is walk out towards the mouth of
the inner ellipse of trees, sight down the line of upper crescent tips, and they
will be looking towards Mecca:

Tower detail with crescent-tip lines. When these lines were
drawn onto the Tower portion of the Crescent site-plan (figure 32 in chapter 4)
the black line pointed directly at Mecca. You can see here that this black line
is the sight line, along the tips of the Tower crescents in the northeast
direction, affording a person coming out from the Tower-sundial a visual
reference for Mecca.

Interestingly, this detail graphic of the Tower array seems
to be rotated clockwise a small amount vis a vis the lower resolution
site-plan graphic of the Tower array seen earlier. In the site-plan image, the
line across the most obtruding crescent tips (red) points 1.8° north of Mecca,
and the line to the further away and more recessed crescent tip (black) points
exactly at Mecca. Here the red line points about 1.4° north of Mecca and the
black line points about 1° south of Mecca.

I don’t know what accounts for these small discrepancies
between the different Tower views. My guess would be that Murdoch decided
(finally) to exercise a modicum of caution, as he seems to have done with his
misleading Tower Section view. There is no reason he had to include the full
Islamofascist precision of his design in the publicly available plans at all. He
could have fudged many details and only put the exact plan into the engineering
drawings.

What is really surprising is how much he did reveal. The
most likely explanation is pride. Why depict the shaft of light passing through
the slot in the tower at just the time of day when a little further westward
movement of the sun will close this shaft off, changing the effective gnomon
height of the tower? If he hadn’t done that, I could not have figured out the
zig-zag in his asr prayer line. But Murdoch had to include this
key information. After all, if the Crescent of Embrace did not win (and what are
the chances that the jury would pick a giant red crescent for the memorial to
Flight 93?), the submitted plans would be all anyone would ever see of Murdoch’s
brilliant creation. He couldn’t leave out the key elements that made the design
a work of genius.

Those who are looking for any possible excuse not to see
the Islamic and jihadist elements of Murdoch’s design might take the lack
of exact consistency in the different views of the crescents of trees that
surround the Tower of Voices as evidence that the Mecca orientations of these
crescents is coincidental. That is about as much of a reed as the deniers have
to hang on: that only one of the two available images of the Tower crescents
shows them to be turned exactly 90° from the central crescent, while the other
shows them turned almost exactly 90°. Maybe a person who focuses hard
enough on that small discrepancy can even avoid seeing the year-round-accurate
Islamic prayer-time sundial Towering overhead.

[1]
From TerraServer’s topo map of Shanksville PA, 7/1/1977 (http://terraserver.microsoft.com/).
There are also some contour lines, without altitude numbers, in the Crescent
site plan. Examination of figure 24 back in chapter 3 reveals a contour line
running pretty much exactly along the inner arc of trees.